According to Alice's frame of reference

According to Alice's frame of reference, there is a 1kg object moving through space at 1 m/s along the x coordinate. When x = 0 meters, Alice applies 1 Newton of force to the object in the positive x direction. However, Bob simultaneously applies 1 Newton of force in the opposite direction.

When the object is at x = 1 meter, and Alice and Bob have each applied 1 Newton to the object (in opposite directions), Alice notes that the object moved 1 meter along the direction that she applied her force of 1 Newton. According to the definition of Work, Work = Force * distance moved in the direction of the force. W = 1N * 1m, which means she transferred 1 Joule of energy to the object.

Yet Alice is perplexed that she sees no evidence of the object's increased energy.

When told that Bob was also applying 1 Newton of force to the object, but in the opposite direction, Alice says "I am not the object's keeper. As far as I know, there are a billion forces acting on that object in every different direction. All I know is that I applied 1 Newton of force to an object that moved 1 meter in the direction of my force, and fulfilled the requirements of the definition of work."

When told that the object was moving in her frame prior to her application of her force, Alice says "The object's energy - including its kinetic energy - prior to my interaction with it is none of my business. If that object was full of gasoline instead of water, I would have increased its energy by just as much with my interaction. I only know how much force I applied to it in the direction that it moved."

Is Alice wrong, and if so why? I have a feeling she is, but I would like it to be explained in the context of work as a relativistic concept (relative of frame and velocity)

Alice may have done one joule of work, but zero work was done on the object, because the work done is equal to the NET force on the object (i.e. vector sum of all the forces acting on it) multiplied by the distance through which it moves. There was ZERO net force on the object.

I don't know how to make relativity factor into my explanation the way you want us to. At 1 m/s, it really isn't a factor anyway.

You didn't mention whether Bob was in the same rest frame as Alice. I guess he must have been, if they were able to agree that they applied the force simultaneously.

I have no idea what the hell Alice's second comment is about. So it had some initial kinetic energy. So what? If some net work was done on it, then it's kinetic energy would have CHANGED by that amount as stated by the work-energy theorem. Since no work was done on it, its kinetic energy didn't change.

Yes, Alice applied a 1N force for a distance of 1 m and so did 1 Nm= 1 Joule of work. Her mistake is in thinking that work will have increased the objects kinetic energy. Work on an object changes its total energy (kinetic plus potential) only in a closed system. The additional force Bob applied (as well as the "billion forces acting on that object in every direction") has to be taken into acount to have a closed system.

I don't know how to make relativity factor into my explanation the way you want us to. At 1 m/s, it really isn't a factor anyway.

Relativity is a basic concept that doesn't only refer to speeds near that of light. http://en.wikipedia.org/wiki/Galilean_relativity My reference to relativity involved differing frames of reference as a possible factor. (A frame of reference that moves along with the object would have definitely seen no net work, not merely because of the two forces simultaneously doing opposing work, but because the object did not move, regardless of forces.)

Staff: Mentor

So although Alice can claim she did work on the object, her noting that the object shows no noticeable signs of increased energy should suggest to her that negative work was simultaneously done on it?

Yes. Ignorance of the law (of conservation of energy) is not an allowable excuse for violating it. She has to accept that she hasn't considered all the energy expended on the object if she wants to reconcile the discrepancy.